An exact analytical solution of the problem on plane elastic bending of the segment of a narrow multilayer beam under the action of arbitrary normal loads distributed over its longitudinal faces is presented. It is assumed that the beam deforms elastically, its layers are made of orthotropic materials homogeneous or continuously heterogeneous across the thickness of layers, which are rigidly connected together, and the load is given as the sum of a trigonometric series. This allowed us to reduce the solution of the given problem on bending to the solution of the auxiliary problem on bending of a multilayer cantilever beam under the action of a sinusoidal load with an arbitrary number of half-waves. Its solution is obtained solving the equations of plane elasticity by using an analytical description for the variables of mechanical characteristics of the multilayer structure. The solution of the original problem is found as the sum of general solutions of the problems for the multilayer cantilever with a load at its free end and with sinusoidal loads on the longitudinal faces. The theoretical relations obtained are checked by solving the test problem on bending of a five-layer hinged beam with a linear discontinuous load. To reduce the Gibbs effect in the vicinity of jump discontinuities of load on its approximation by the partial sum of a trigonometric series, the Lancos method was used. The results obtained are confirmed by the results of a finite-element modeling. The solution constructed enables one to take into account an arbitrary distribution of normal load on longitudinal beam faces, including local loads and loads on a surface section, and can be used for predicting the strength and stiffness of multilayer beams and, with small changes — for solving contact problems for such structural elements.
Similar content being viewed by others
References
A. S. Sayyad and Y. M. Ghugal, “Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature,” Compos. Struct., 171, 486-504 (2017).
V. G. Piskunov, A. V. Goryk, and V. N. Cherednikov, “Modeling of transverse shears of piecewise homogeneous composite bars using an iterative process with account of tangential loads. 1. Construction of a model,” Mech. Compos. Mater., 36, No. 4, 287-296 (2000).
V. I. Shvabyuk, S. V. Rotko, and O. A. Uzhegova, “Bending of a Composite Beam with a Longitudinal Section,” Strenght of Materials., 46, No. 4, 558-566 (2014).
S. G. Lekhnitskii, Anisotropic Plates, N. Y., Gordon and Breach, (1968).
R. W. Gerstner, “Stresses in a composite cantilever,” J. Compos. Mater., 2, No. 4, 498-501 (1968).
S. Cheng, X. Wei, T. Jiang, “Stress distribution and deformation of adhesive-bonded laminated composite beams,” ASCE J. Eng. Mech., 115, 1150-1162 (1989).
L. Zhao, W.Q. Chen, and C. F. Lü, “New assessment on the Saint-Venant solutions for functionally graded beams,” Mech. Res. Comm., 43, 1-6 (2012).
A. V. Goryk and S. B. Kovalchuk, “Elasticity theory solution of the problem on plane bending of a narrow layered cantilever bar by loads at its end,” Mech. Compos. Mater., 54, No. 2, 179-190 (2018).
A.-M. Jiang and H.-J. Ding, “The analytical solutions for orthotropic cantilever beams (I): Subjected to surface forces,” J. Zhejiang Univ.: Sci. A., 6, No. 2, 126-131 (2005).
H. J. Ding, D. J. Huang, and H. M. Wang, “Analytical solution for fixed-fixed anisotropic beam subjected to uniform load,” Appl. Math. Mech., 27, No. 10, 1305-1310 (2006).
D.-J. Huang, H.-J. Ding, and W.-Q. Chen, “Analytical solution for functionally graded anisotropic cantilever beam under thermal and uniformly distributed load,” J. Zhejiang Univ.: Sci. A., 8, No. 9, 1351-1355 (2007).
Z. Zhong, T. Yu, “Analytical solution of a cantilever functionally graded beam,” Composites Science and Technology, 67, No 3-4, 481-488 (2007).
M. Wang and Y. Liu, “Analytical solution for bi-material beam with graded intermediate layer,” Compos. Struct., 92, 2358-2368 (2010).
A. Daneshmehr, S. Momeni, and M. R. Akhloumadi, “Exact elasticity solution for the density functionally gradient beam by using airy stress function,” Appl. Mech. Mater., 110-116, 4669-4676 (2012).
Q. Yang, B. L. Zheng, K. Zhang, and J. Li, “Elastic solutions of a functionally graded cantilever beam with different modulus in tension and compression under bending loads,” Appl. Math. Modeling., 38, No. 4, 1403-1416 (2014).
S. Benguediab, A. Tounsi, H. H. Abdelaziz, and M. A. A. Meziane, “Elasticity solution for a cantilever beam with exponentially varying properties,” J. Appl. Mech. Technical Phys., 58, No. 2, 354-361 (2017).
A. V. Goryk and S. B. Koval’chuk, “Solution of a transverse plane bending problem of a laminated cantilever beam under the action of a normal uniform load,” Strength of Materials., 50, No. 3, 406-418 (2018).
S. Koval’chuk and A. V. Goryk, “Exact solution of the problem of elastic bending of a multilayer under the action of a normal uniform load,” Mater. Sci. Forum, 968, 475-485 (2019).
U. Esendemir, M. R. Usal, and M. Usal, “The effects of shear on the deflection of simply supported composite beam loaded linearly,” J. Reinf. Plast. Compos., 25, 835-846 (2006).
D.-J. Huang, H.-J. Ding, and W.-Q. Chen, “Analytical solution for functionally graded anisotropic cantilever beam subjected to linearly distributed load,” Applied Mathematics and Mechanics, 28, No. 7, 855-860 (2007).
T. H. Daouadji, A. H. Henni, A. Tounsi, and A. B. El Abbes, “Elasticity solution of a cantilever functionally graded beam,” Appl. Compos. Mater., 20, No. 1, 1-15 (2013).
N. J. Pagano, “Exact solutions for composite laminates in cylindrical bending,” J. Compos. Mater., 3, 398-411 (1969).
B. V. Sankar, “An elasticity solution for functionally graded beams,” J. Compos. Sci. Technol., 61, No. 5, 689-696 (2001).
I. K. Silverman, “Orthotropic beams under polynomial loads,” ASCE J. Eng. Mech. Div., 90, 293-319 (1964).
Z. Hashin, “Plane anisotropic beams,” J. Appl. Mech., 34, No. 2, 257-262 (1967).
A. M. Jiang and H. J. Ding, “The analytical solutions for orthotropic cantilever beams (II): solutions for density functionally graded beams,” J. Zhejiang Univ.: Sci. A., 6, No. 3, 155-158 (2005).
H. J. Ding, D. J. Huang, and W. Q. Chen, “Elasticity solutions for plane anisotropic functionally graded beams,” Int. J. Solids and Struct., 44, No. 1, 176-196 (2007).
D.-J. Huang, H.-J. Ding, and W.-Q. Chen, “Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading,” Sci. China. Ser. G: Physics, Mechanics and Astronomy., 52, No. 8, 1244-1256 (2009).
G. J. Nie, Z. Zhong, and S., “Chen Analytical solution for a functionally graded beam with arbitrary graded material properties,” Composites: Part B., 44, 274-282 (2013).
L. Zhang, P. Gao, and D. Li, “New methodology to obtain exact solutions of orthotropic plane beam subjected to arbitrary loads,” J. Eng. Mech., 138, No. 11, 1348-1356 (2012).
S. B. Koval’chuk and A. V. Goryk, “Elasticity theory solution of the problem on bending of a narrow multilayer cantilever with a circular axis by loads at its end,” Mech. Compos. Mater., 54, No. 5, 605-620 (2018).
M. M. Filonenko-Borodich, Elasticity Theory [in Russian], M., Gos. Izd. Fiz. Mat. Lit. (1959).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Mekhanika Kompozitnykh Materialov, Vol. 56, No. 1, pp. 81-108, January-February, 2020.
Rights and permissions
About this article
Cite this article
Koval’chuk, S.B. Exact Solution of the Problem on Elastic Bending of the Segment of a Narrow Multilayer Beam by an Arbitrary Normal Load. Mech Compos Mater 56, 55–74 (2020). https://doi.org/10.1007/s11029-020-09860-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11029-020-09860-y